Exercises 27–40 contain linear equations with constants in denominators. Solve each equation. x/3 = x/2 - 2

Verified step by step guidance

1. Identify the equation: \(\frac{x}{3} = \frac{x}{2} - 2\).
2. Find the least common denominator (LCD) of the denominators 3 and 2, which is 6.
3. Multiply every term in the equation by the LCD (6) to eliminate the denominators: \$6 \times \frac{x}{3} = 6 \times \frac{x}{2} - 6 \times 2$.
4. Simplify each term after multiplication: \$2x = 3x - 12$.
5. Isolate the variable \(x\) by subtracting \$3x\( from both sides: \)2x - 3x = -12\(, then simplify to get \)-x = -12$.

Key Concepts

Solving Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Solving such equations involves isolating the variable on one side to find its value. Techniques include combining like terms and performing inverse operations.

Clearing Fractions by Multiplying Both Sides

When an equation contains fractions, multiplying both sides by the least common denominator (LCD) eliminates the denominators, simplifying the equation. This step helps avoid dealing with fractions and makes solving the equation more straightforward.

Checking for Extraneous Solutions

After solving an equation, especially those involving variables in denominators, it is important to check solutions by substituting them back into the original equation. This ensures no solution makes a denominator zero, which would be undefined and thus extraneous.